New bounds on the cardinality of Hausdorff spaces and regular spaces

Using weaker versions of the cardinal function $$\psi_c(X)$$ , we derive a series new bounds for cardinality Hausdorff spaces and regular that do not involve nor its variants at all. For example, show if X is then $$|X|\leq 2^{c(X)^{\pi\chi(X)}}$$ 2^{c(X)\pi\chi(X)^{ot(X)}}$$ where $$ot(X)$$ introduced by Tkachenko, has property $$ot(X)\leq\min\{t(X),c(X)\}$$ . It follows from latter space with cellularity most $$\mathfrak{c}$$ countable $$\pi$$ -character $$2^\mathfrak{c}$$ 2^{d(X)^{\pi\chi(X)}}, \ |X|\leq d(X)^{\pi\chi(X)^{ot(X)}}, \text { } 2^{\pi w(X)^{dot(X)}}, $$ dot(X)\leq\min\{ot(X),\pi\chi(X)\}$$ None these or $$\psi(X)$$ By introducing functions $$w\psi_c(X)$$ $$d\psi_c(X)$$ $$w\psi_c(X)d\psi_c(X)\leq\psi_c(X)$$ X, $$|X|\leq\pi\chi(X)^{c(X)w\psi_c(X)}$$ $$|X|\leq\pi\chi(X)^{c(X)d\psi_c(X)w\psi_c(X)}$$ Hausdorff. This improves results Šapirovskiĭ Sun. also shown 2^{d(X)w\psi_c(X)}$$ which appears to be even in case replaced Compact examples cannot $$d\psi_c(X)w\psi_c(X)$$ bound $$2^{\psi(X)}$$ compact X. Likewise, Arhangel׳skiĭ-Šapirovskiĭ $$2^{L(X)t(X)\psi(X)}$$ Finally, make several observations concerning homogeneous this connection.